1. Thesis Defense Exam Presentation
Development of Fuzzy Syllogistic Algorithms and
Applications Distributed Reasoning Approaches
Hüseyin Çakır
Izmir Institude of Technology
huseyincakir@iyte.edu.tr

2. Contents
●
Introduction
●
Research Approach
●
Background
●
Structural Analysis of Syllogisms
●
Applications for Syllogistic Reasoning
●
Conclusion
15/12/10
2

3. Introduction
●
A syllogism is a logical argument in which
conclusion can be inferred from two other
premises.
Example:
ALL PRIMATES ARE MAMMALS <<major premiss>>
ALL HUMANS ARE PRIMATES <<minor premiss>>
--------------------------------------------ALL HUMANS ARE MAMMALS <<conclusion>>
15/12/10
3

4. Introduction
●
The aim of the thesis was to:
●
Use syllogisms as reasoning mechanism.
●
Analyze the structural properties of syllogisms.
●
●
●
15/12/10
Introduce the fuzzy syllogisms, which helps giving
possibilistic values to syllogistic propositions.
Verify the truth of the approach with applications.
Discuss the possibble application areas and
drawbacks of syllogistic reasoning.
4

5. Introduction
●
●
Computational logic can be used to model
syllogistic reasoning, originally developed by
Aristotle some 2.300 years ago.
By modelling syllogisms, it is possibble to
analyze the stuctural properties of syllogisms
and syllogistic search space.
15/12/10
5

6. Research Approach
●
Aim of the thesis
●
Literature survey
●
Development
●
Application
●
Conclusion
15/12/10
6

7. Background/ Syllogism
●
●
The origin of the logic studies known goes
among ancient Babylonian, Greeks, Indian,
Chiese and Islamic cultures.
Aristotle's theory suggests that in some cases
the answer (conclusion) is predictable based on
earlier answers which called premisses.
Example:
ALL PRIMATES ARE MAMMALS <<major premiss>>
ALL HUMANS ARE PRIMATES <<minor premiss>>
--------------------------------------------ALL HUMANS ARE MAMMALS <<conclusion>>
15/12/10
7

8. Background/ Syllogism
●
Depending on alternative placements of the
objects within the premises, 4 basic types of
syllogistic figures are possible.
Figure Name
I
II
III
IV
Major Premise
Minor Premise
――――――
Conclusion
MP
SM
――
SP
PM
SM
――
SP
MP
MS
――
SP
PM
MS
――
SP
Example: Figure 1 MAMMALS: MAJOR HUMANS: MINOR PRIMATES: MIDDLE
ALL PRIMATES ARE MAMMALS
ALL M ARE P
ALL HUMANS ARE PRIMATES
ALL S ARE M
----------------------------------------------------------------------------------------ALL HUMANS ARE MAMMALS
ALL S ARE P
15/12/10
8

9. Background/ Syllogism
●
Propositions has a number of dualistic
attributes that characterize the propositions.
Name
Universality
Positivity
A
Universal
positive
E
Universal
negative
I
Particular
positive
O
Particular
negative
Example: Figure 1 - AAA
ALL PRIMATES ARE MAMMALS
ALL HUMANS ARE PRIMATES
--------------------------------------------ALL HUMANS ARE MAMMALS
15/12/10
ALL M ARE P
ALL S ARE M
--------------------------------------------ALL S ARE P
9

10. Background/ Syllogism
●
The letters A, E, I, O have been used since the
medieval schools and to memorise valid moods
mnemonic names used as follows:
Figure 1
Figure 2
Figure 3
Figure 4
Barbara (AAA)
Cesare (EAE)
Datisi (AII)
Calemes (AEE)
Celarent (EAE)
Camestres
(AEE)
Disamis (IAI)
Dimatis (IAI)
Darii (AII)
Festino (EIO)
Ferison (EIO)
Fresison (EIO)
Ferio (EIO)
Baroco (AOO)
Bocardo (OAO)
Calemos (AEO)
Barbari (AAI)
Cesaro (EAO)
Felapton (EAO)
Fesapo (EAO)
Celaront (EAO)
Camestros
(AEO)
Darapti (AAI)
Bamalip (AAI)
15/12/10
10

11. Background/ Syllogism
●
●
Aristotle had specified the first three figures.
The 4th figure was discovered in the middle
age.
The first proposition consist of a quantified
relationship between the objects M and P, the
second proposition of S and M, the conclusion
of S and P.
Figure Name
II
III
IV
Major Premise
Minor Premise
――――――
Conclusion
15/12/10
I
MP
SM
――
SP
PM
SM
――
SP
MP
MS
――
SP
PM
MS
――
SP
11

12. Background/ Syllogism
Figure Name
II
III
IV
Major Premise
Minor Premise
――――――
Conclusion
15/12/10
I
MP
SM
――
SP
PM
SM
――
SP
MP
MS
――
SP
PM
MS
――
SP
12

13. Background/ Syllogism
●
Since the proposition operator may have 4
values, 64  syllogistic moods are possible for
every figure and 256 moods for all 4 figures in
total.
FIGURE I
FIGURE III
FIGURE IV
AAA -1
AAO -1
AAE - 1
AAI - 1
...
15/12/10
FIGURE II
AAA - 2
AAO - 2
AAE - 2
AAI - 2
...
AAA - 3
AAO - 3
AAE - 3
AAI - 3
...
AAA - 4
AAO - 4
AAE - 4
AAI - 4
...
13

14. Background/ Syllogism
●
Invalid syllogisms are also one of the most
important issue of syllogisms.
●
Affirmative conclusion from a negative premise.
–
●
Existential fallacy.
–
●
Conclusion I or O while premiss is E or A.[Ex: AAI]
Fallacy of exclusive premises.
–
15/12/10
Conclusion A or I while premiss is E or O.[Ex: AEA]
Two negative premisses. [Ex: EEA]
14

15. Background/ Syllogism
●
Fallacy of the undistributed middle.
–
●
Illicit major/minor.
–
●
Middle term must be distributed in at least one premiss.
No term can be distributed in conclusion which is not
distributed in premiss.
Fallacy of necessity.
–
Exactly three terms, used in same sense.
Statement
Subject P
ALL M ARE P (A)
Disributed
Undistributed
ALL M ARE NOT P (E)
Distributed
Distributed
SOME M ARE P (I)
15/12/10
Subject M
Undistributed
Undistributed
SOME M ARE NOT P
(O)
Undistributed
Distributed
15

16. Background/ Reasoning
●
The syllogism is part of deductive reasoning,
where facts are determined by combining
existing statements, in contrast to inductive
reasoning.
15/12/10
16

17. Background/ Formal Representation
●
Formal representation of syllogisms can be
made by using several approaches:
●
Euler Diagram Representation
●
Venn Diagram Representation
●
Linear Representation
●
...
15/12/10
17

18. Background/ Formal Representation
●
The terms in a proposition are related to each
other in four different ways. (Set-Theoretic App.)
Operator 
Proposition 
A
All S are P
E
All S are not P
I
Some S are P
O
Some S are not P
15/12/10
Set-Theoretic Representation of Logical Cases
18

19. Background/ Fuzzy Logic
●
●
Fuzzy logic is reasoning that is approximate
rather than accurate. (opposite of crisp logic)
Fuzzy logic variables can have a truth value
that ranges between 0 and 1.
Possibility
Probability
15/12/10
19

20. Background/ Application Areas
●
Data mining
●
Object-oriented programming
●
Semantic Web
●
Artificial Intelligence/ Reasoning
15/12/10
20

21. Structural Analysis of Syllogisms
●
●
For three symmetrically intersecting sets there
are in total 11 possible sub-sets in a Venn
diagram.
If symmetric set relationships are relaxed and
the three sets are named, for instance with the
syllogistic terms P, M and S, then 41 set
relationships are possible.
15/12/10
21

22. Structural Analysis of Syllogisms
Example:
11
distinct set
situations
...
...
15/12/10
41
Set
relationships
22

23. Structural Analysis of Syllogisms
M
P
g
f
a+e
a+b
a+c
d
S
15/12/10
23

24. Structural Analysis of Syllogisms
●
●
9 distinct relationships exists between the three
sets P, M and S.
For instance P∩M is mapped onto 1=a+e and
P-M is mapped onto 4=f+b.
Sub-Set Number 1
2
3
4
5
6
7
8
9
Arithmetic
Relation
a+e
a+c
a+b
f+b
f+e
g+c
g+e
d+b
d+c
Syllogistic Case
P∩M M∩S S∩P
P-M
P-S
M-P
M-S
S-M
S-P
15/12/10
24

25. Structural Analysis of Syllogisms
Sub-Set Number 1
2
3
4
5
6
7
8
9
Arithmetic
Relation
a+e
a+c
a+b
f+b
f+e
g+c
g+e
d+b
d+c
Syllogistic Case
P∩M M∩S S∩P
P-M
P-S
M-P
M-S
S-M
S-P
#21
1
0
1
1
1
1
1
15/12/10
0
0
25

26. Structural Analysis of Syllogisms
Example: Figure 1 - AAA
ALL PRIMATES ARE MAMMALS
ALL HUMANS ARE PRIMATES
--------------------------------------------ALL HUMANS ARE MAMMALS
ALL M ARE P
ALL S ARE M
--------------------------------------------ALL S ARE P
Sub-Set Number 1
2
3
4
5
6
7
8
9
Arithmetic
Relation
a+e
a+c
a+b
f+b
f+e
g+c
g+e
d+b
d+c
Syllogistic Case
P∩M M∩S S∩P
P-M
P-S
M-P
M-S
S-M
S-P
0
0
0
15/12/10
26

27. Structural Analysis of Syllogisms
●
Valid Stiuations:
Sub-Set Number 1
2
3
4
5
6
7
8
9
Arithmetic
Relation
a+e
a+c
a+b
f+b
f+e
g+c
g+e
d+b
d+c
Syllogistic Case
P∩M M∩S S∩P
P-M
P-S
M-P
M-S
S-M
S-P
#25
1
1
1
0
1
0
0
15/12/10
1
1
27

28. Structural Analysis of Syllogisms
●
The above homomorphism represents the
essential data structure of the algorithm for
deciding syllogistic moods.
Arithmetic
Relation
a+e
a+c
a+b
f+b
f+e
g+c
g+e
d+b
d+c
#1
1
1
1
1
1
0
1
0
0
...
...
...
...
...
...
...
...
...
#2
...
#41
15/12/10
28

29. Structural Analysis of Syllogisms
●
The pseudo code of the algorithm for
determining the true and false cases of a given
moods is based on selecting the possible set
relationships for that mood, out of all 41
possible set relationships.
15/12/10
29

30. Structural Analysis of Syllogisms
Pseudocode:
DETERMINE mood
READ figure number {1,2,3,4}
READ with 3 proposition ids {A,E,I,O}
GENERATE 41 possible set combinations with 9 relationships into an array
SetCombi[41,9]={{1,1,1,1,1,1,1,1,1}, ..., {0,1,0,0,1,1,1,1,1}}
VALIDATE every proposition with either validateAllAre,
validateAllAreNot, validateSomeAreNot or validateSomeAre
DISPLAY valid and invalid cases of the mood
VALIDATE mood
validateAllAre(x,y) //all M are P
if(x=='M' && y=='P')
CHECK the sets suitable for this mood in setCombi
if 1=1 and 2=0 then add this situation as valid
if(setCombi[i][0]==1 && setCombi[i][1]==0)
//similar for validateAllAreNot(), validateSomeAre(),validateSomeAreNot()
15/12/10
30

31. DETERMINE MOOD
Structural Analysis of Syllogisms
FIGURE 1,2,3,4
PROPOSITION A,E,I,O
GENERATE 41 POSSIBLE SET COMBINATIONS
SET RELATIONSHIPS INTO ARRAY
VALIDATE EVERY PROPOSITION
15/12/10
31

32. Structural Analysis of Syllogisms
●
●
Statistics gained from the algorithm mentioned
in previous section.
This algorithm provides some beneficial
statistics about syllogisms which enables
understanding the structural behaviours of
syllogisms.
15/12/10
32

33. Structural Analysis of Syllogisms
●
●
According to the model there exists 11 distinct
relations among Venn Diagrams that provide
determining syllogisms.
Every mood has 0 to 21 true and 0 to 21 false
cases, which is a real subset of the 41 distinct
cases.
15/12/10
33

34. Structural Analysis of Syllogisms
●
●
For any given figure the total number of all true
cases is equal to all false cases, ie 328 true and
328 false cases.
For all 4 syllogistic figures the total number of 4
x 2 x 328 = 2624 cases.
15/12/10
34

35. Structural Analysis of Syllogisms
MOOD
# of valids
# of invalids
valid cases
------------------------------------------------------------------mood[2]:
|
0
|
1
|
mood[4]:
|
0
|
1
|
mood[10]:
|
0
|
6
|
mood[17]:
|
0
|
1
|
mood[19]:
|
0
|
1
|
mood[25]:
|
0
|
7
|
mood[1]:
|
1
|
0
|-25mood[3]:
|
1
|
0
|-25mood[5]:
|
1
|
2
|-29mood[6]:
|
1
|
2
|-21mood[14]:
|
1
|
7
|-21mood[49]:
|
2
|
6
|-5—10…
------------------------------------------------------------------TOTAL NUMBER OF VALID SUBSETS FOR THIS FIGURE:328
TOTAL NUMBER OF INVALID SUBSETS FOR THIS FIGURE:328
TOTAL NUMBER OF SUBSETS FOR THIS FIGURE:656
-------------------------------------------------------------------
15/12/10
35

36. Structural Analysis of Syllogisms
22
21
20
19
18
17
16
15
14
13
12
11
valid
invalid
10
9
8
7
6
5
4
3
2
1
0
1
0
15/12/10
3
2
5
4
7
6
9
8
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65
10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64
36

37. Structural Analysis of Syllogisms
●
Reducing fallacies:
Rule 1, “convert E into O since the information in O
also contains the information in E”.
Rule 2 , “convert A into I since the information in A
also contains the information in I”.
15/12/10
37

38. 15/12/10
Valids for Figure 1:
Mood[1]:
AAA
mood[3]:
AAI
Mood[11]:
AII
mood[18]:
EAE
Mood[20]:
EAO
Mood[28]:
EIO
Mood[26]:
*EIE
Mood[9]:
*AIA
mood[20]:
mood[11]:
mood[1]:
mood[24]:
mood[19]:
mood[15]:
mood[7]:
mood[2]:
mood[52]:
mood[35]:
mood[5]:
mood[60]:
mood[51]:
mood[36]:
mood[27]:
mood[21]:
mood[12]:
mood[43]:
mood[26]:
mood[50]:
mood[47]:
mood[33]:
mood[53]:
mood[44]:
mood[10]:
mood[59]:
mood[25]:
mood[29]:
mood[58]:
mood[45]:
●
mood[42]:
mood[57]:
Structural Analysis of Syllogisms
Change in conclusion: (Figure 1)
25
20
15
10
valid
invalid
valid
5
invalid
0
38

39. 15/12/10
Valids for Figure 2:
Mood[6]:
AEE
Mood[8]:
AEO
Mood[16]:
AOO
Mood[18]:
EAE
Mood[20]:
EAO
Mood[28]:
EIO
Mood[14]:
*AOE
Mood[26]:
*EIE
mood[20]:
mood[16]:
mood[6]:
mood[24]:
mood[19]:
mood[12]:
mood[5]:
mood[3]:
mood[51]:
mood[35]:
mood[11]:
mood[55]:
mood[49]:
mood[36]:
mood[27]:
mood[21]:
mood[2]:
mood[43]:
mood[26]:
mood[59]:
mood[47]:
mood[33]:
mood[64]:
mood[44]:
mood[10]:
mood[53]:
mood[25]:
mood[29]:
mood[57]:
mood[45]:
●
mood[42]:
mood[61]:
Structural Analysis of Syllogisms
Change in conclusion: (Figure 2)
25
20
15
10
valid
invalid
valid
5
invalid
0
39

40. 15/12/10
Valids for Figure 3:
Mood[3]:
AAI
Mood[11]:
AII
Mood[20]:
EAO
Mood[28]:
EIO
Mood[35]:
IAI
Mood[52]:
OAO
Mood[1]:
*AAA
Mood[9]:
*AIA
Mood[18]:
*EAE
Mood[26]:
*EIE
Mood[33]:
*IAA
Mood[50]:
*OAE
mood[35]:
mood[20]:
mood[3]:
mood[36]:
mood[24]:
mood[19]:
mood[8]:
mood[4]:
mood[40]:
mood[6]:
mood[1]:
mood[55]:
mood[31]:
mood[22]:
mood[17]:
mood[13]:
mood[9]:
mood[43]:
mood[26]:
mood[54]:
mood[37]:
mood[30]:
mood[53]:
mood[44]:
mood[10]:
mood[48]:
mood[29]:
mood[63]:
mood[58]:
mood[45]:
●
mood[42]:
mood[57]:
Structural Analysis of Syllogisms
Change in conclusion: (Figure 3)
25
20
15
10
valid
invalid
5
valid
invalid
0
40

41. 15/12/10
Valids for Figure 4:
Mood[3]:
AAI
Mood[4]:
AAO
Mood[6]:
AEE
Mood[8]:
AEO
Mood[20]:
EAO
Mood[28]:
EIO
Mood[35]:
IAI
Mood[1]:
*AAA
Mood[2]:
*AAE
Mood[18]:
*EAE
Mood[26]:
*EIE
Mood[33]:
*IAA
mood[28]:
mood[8]:
mood[4]:
mood[52]:
mood[36]:
mood[24]:
mood[19]:
mood[12]:
mood[5]:
mood[1]:
mood[18]:
mood[56]:
mood[39]:
mood[27]:
mood[21]:
mood[15]:
mood[38]:
mood[60]:
mood[37]:
mood[30]:
mood[44]:
mood[10]:
mood[63]:
mood[53]:
mood[49]:
mood[47]:
mood[25]:
mood[9]:
mood[61]:
●
mood[57]:
mood[45]:
mood[42]:
Structural Analysis of Syllogisms
Change in conclusion: (Figure 4)
20
18
16
14
12
10
8
valid
6
invalid
4
valid
2
invalid
0
41

42. Structural Analysis of Syllogisms
●
Fuzzy Syllogisms:
●
●
The results discussed above used same approach
as in Aristotle 's, so it decides on syllogisms as valid
or invalid which gives strict decisions on syllogisms
either name them as true or false.
But our objective is to utilize the full set of all
256 moods as a fuzzy syllogistic system of
possibilistic arguments.
15/12/10
42

43. Structural Analysis of Syllogism
●
The truth values for every mood in form of a
truth ration between its true and false cases, so
that the truth ratio becomes a real number,
normalized within [0, 1].
15/12/10
43

44. Structural Analysis of Syllogism
15/12/10
44

45. Structural Analysis of Syllogism
15/12/10
45

46. Structural Analysis of Syllogism
●
Certainly Not:
8
7
6
5
4
INVALID
VALID
3
2
1
0
EIA - 2
EIA - 1
EIA - 4
EIA - 3
15/12/10
AIE - 3
AIE - 1
OAA - 3
AOA - 2
EAA - 3
AAE - 1
EAA - 1
AEA - 2
EAA - 2
AAA - 4
AEA - 4
IAE - 3
IAE - 4
AAE - 3
EAA - 4
AAO - 1
EAI - 1
AEI - 2
EAI - 2
AAE - 4
AEI - 4
46

47. Structural Analysis of Syllogism
●
Unlikely:
25
20
15
INVALID
10
VALID
5
0
OOA - 1
IIE - 4
IOA - 3
IIA - 2
IOE - 2
EOA - 2 OEE - 4 EOA - 4
IAE - 2
OEE - 1
IEA - 1
OAE - 3
EIE - 3
OAE - 2 EEE - 2
AEE - 1
OOA - 2
IIE - 1
OOE - 1 IOA - 4
IOA - 1
OIA - 4 OEA - 2 EOA - 3 OAA - 1 AOA - 1 OEE - 3
IEA - 4
IEE - 1
IEE - 4
EAA - 1 EEA - 4 AEE - 3
15/12/10
47

48. Structural Analysis of Syllogism
Uncertain:
●
3,5
3
2,5
2
1,5
1
0,5
0
AIA - 1
15/12/10
AIO - 1
AIA - 3
AIO - 3
AOA - 3
AOO - 3
48

49. Structural Analysis of Syllogism
Likely:
●
25
20
15
INVALID
10
VALID
5
0
OOO - 3
III - 4
IOO - 2
OOI - 1
OOI - 4 EOO - 3 OAO - 4 OEO - 4
IAI - 2
IEO - 1
OAI - 1
EOI - 4
IEI - 3
EEO - 2 EEO - 1 AAO - 3
OIO - 1
III - 1
OII - 2
IIO - 3
OII - 3
EOO - 1 AOO - 4 OEI - 4 OEO - 1 IAO - 4
IEO - 4
EOI - 3
EII - 2
AAI - 2
EEI - 4 OAO - 2 EAI - 3
15/12/10
49

50. Structural Analysis of Syllogism
●
Certainly
8
7
6
5
4
INVALID
VALID
3
2
1
0
EIO - 2
EIO - 1
15/12/10
EIO - 4
EIO - 3
AII - 3
AII - 1
OAO - 3
IAI - 3
IAI - 4
AOO - 2
EAO - 3
AAA - 1
EAE - 1
AEE - 2
EAE - 2
AAI - 4
AEE - 4
AAI - 3
EAO - 4
AAI - 1
EAO - 1
AEO - 2
EAO - 2
AAO - 4
AEO - 4
50

51. Applications for Syllogistic
Reasoning
●
During this study various applications
developed to check validty of algorithm.
●
●
●
15/12/10
Mathematical applications to check validity of
algorithm and to reveal statistics about syllogism.
Application that use syllogistic reasoning in
distributed way.
Use of syllogistic reasoning in object-oriented
programming.
51

52. Applications for Syllogistic
Reasoning
●
Application 1: Listing all valid/invalid set
situations.
MOOD
# of valids
# of invalids
valid cases
------------------------------------------------------------------mood[2]:
|
0
|
1
|
mood[4]:
|
0
|
1
|
mood[10]:
|
0
|
6
|
mood[17]:
|
0
|
1
|
mood[19]:
|
0
|
1
|
mood[25]:
|
0
|
7
|
mood[1]:
|
1
|
0
|-25mood[3]:
|
1
|
0
|-25mood[5]:
|
1
|
2
|-29mood[6]:
|
1
|
2
|-21mood[14]:
|
1
|
7
|-21...
15/12/10
52

53. Applications for Syllogistic
Reasoning
●
Application 2:
15/12/10
53

54. Applications for Syllogistic
Reasoning
●
Application 3:
15/12/10
54

55. Applications for Syllogistic
Reasoning
●
Application 4:
15/12/10
55

56. Applications for Syllogistic
Reasoning
●
Application 5:
15/12/10
56

57. Conclusion
●
●
Mathematical properties of the whole syllogistic
system are revealed in detail including
applications and statistics.
It is believed that this thesis has two
contributions to the literature, specifically to the
search space of syllogisms and to the
fuzzification of syllogistic values.
15/12/10
57

58. Conclusion
●
●
The principles that have been developed in this
thesis work can be used as a reference in
developing some applications about syllogistic
reasoning.
The reason why it contributes to syllogistic
reasoning field is that it shows the whole validity
values for all moods in all figures.
15/12/10
58

59. Conclusion
●
●
A computer software, that provides the
necessary aid to the programmer as software
editor can also be developed as a future work.
This will enable the syllogistic reasoning used in
applications which will make remarkable
contribution to syllogistic reasoning approach.
15/12/10
59